What are some quotes from Homer's epic poem that show that the Greeks lived in a patriarchal society?
Homer's epic poem Odyssey emerged from a patriarchal society, i.e., a society in which the father held the dominant role in a family. This is not unusual for cultures around the Mediterranean. The same can be said of the ancient Romans and Hebrews, for example.
Such patriarchy can be seen easily in the ruler of the Greek gods, Zeus, whom Homer frequently calls "The Father of gods and men" (A.S. Kline translation).
It is also interesting to note that Athene tells Telemachus that if Penelope wants to marry, then she should "go back to her great father’s house" (A.S. Kline translation), which indicates that even as a fully mature woman with a son, Penelope is still, to some extent, under her father's control (assuming that Odysseus was deceased).
We even see that in Odysseus' absence, his son Telemachus seems to have some measure of control over his mother. In Odyssey 1, when Penelope does not want to listen to the sorrowful song of Phemius, Telemachus rebukes her and says:
"So go to your quarters now, and attend to your own duties at loom and spindle, and order your maids about their tasks: let men worry about such things, and I especially, since I hold the authority in this house" (A.S. Kline translation).
Thus, it is clear that the Greek society as Homer portrays it in the Odyssey is a patriarchal one. We should not forget, however, that women play important roles in this epic. Odysseus and his family benefitted greatly from the help of Athena, and Homer views Penelope's wisdom in a very favorable light.